{"id":"8ebaf53e-700e-4c2a-9460-cbaf9d3e7616","arxiv_id":"2607.14022","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"New supersaturation theorems for hypergraph-weighted independent sets give new bounds for generalized Turán problems and for systems of equations in integers.","lead":"This paper proves a general mathematical theorem: once a finite structure is forced to contain a forbidden pattern, it often must contain many copies of that pattern at once. The authors use this framework to get new counting bounds for graphs and for sets of integers, extending recent work by Ferber, McKinley and Samotij.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 4.8 proof gap: |I_t|<|t| does not imply |I_t|≤|t|-2; types with a single vertex in a non-singleton part break the second bound of Theorem 1.7.","rationale":"The reader's conditional verdict is appropriate, but the stated weakest assumption—the no-isolated-vertices condition in Theorem 1.1—is an explicit and correctly identified limitation of the application, not a hidden flaw in the central mechanism. The more load-bearing issue I found is in the proof of the central Theorem 1.7 itself: Proposition 4.8's Claim 4.10 contains a false combinatorial assertion about the size of I_t. This is not a matter of disputed conventions or prior work; it is an internal step. The second bound of Theorem 1.7 is explicitly acknowledged as non-natural and is used for several applications, so a gap there affects the main technical engine. My concrete test targets exactly the smallest configuration where the false assertion fails, and asks whether the theorem's hypothesis ex(n,P)=O(n^β) supplies the missing control. I do not claim the theorem is false; I claim the proof as written is incomplete at a specific, checkable point. The reader's verdict remains CONDITIONAL: the paper should be accepted only after this step is repaired or independently verified.","tokens_in":21585,"tokens_out":49486,"duration_ms":435183,"concrete_test":"Construct the minimal example: h=3, parts A={a1,a2}, B={b1,b2}, C={c1,c2}; let H have edges {a1,b1,c1} and {a2,b2,c2}, and F have the single edge {a1,b2}. This pair satisfies the no-containment condition and has type t=(1,1,0) with |I_t|=1=|t|-1. Apply Proposition 4.5 with shrinking chosen so that A becomes {a1} while B remains size 2, then explicitly evaluate inequality (6) of Claim 4.10. The resulting bound should be roughly C k^{-1} p^{-(h−β−2)}, which cannot be made smaller than c^{-1} k^{-2/(h−β−|t|+2)} for large k. Then either exhibit a hereditary family containing this pair with ex(n,P)=O(n^β) and arbitrarily large k, showing Theorem 1.7's second bound is false, or identify the additional use of ex(n,P) that repairs the argument.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.7's second bound rests on Proposition 4.8. In Claim 4.10, the proof must show for every relevant type t that inequality (6) holds. In the case |t|<h−β, it asserts: 'if |I_t|<|t| then |I_t|≤|t|−2 since such a t has some t_i≥2.' This assertion is false. I_t only counts indices with |V_i'|=1 and t_i=1. An F-edge of type t=(1,1,0) in a 3-partite H with one used part not shrunk to a singleton has |t|=2 and |I_t|=1, with no t_i≥2. Such a type is compatible with the no-containment hypothesis: the H-edge involving that non-singleton part can use a different vertex in the first part. Repeating the estimate (7) for this situation gives an upper bound of order k^{-1/(h−β−|t|+1)} rather than k^{-2/(h−β−|t|+2)}; for large k this is too weak to force e_t(F')<1. Thus the written proof does not establish e_t(F)=Ω(k^{2/(h−β−|t|+2)}) for such types. Since Claim 4.10 must hold for all types to conclude F'=0, the second part of Theorem 1.7—and hence Theorems 1.1(iii), 1.3, and 1.4(iii)—is not fully proved as written.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a general framework for supersaturation in which one seeks subsets that are independent in a hypergraph F while inducing many edges in another hypergraph H on the same vertex set, encoded by the quantity α_H(F). The main abstract results are Theorem 1.5, proved by a randomized deletion argument in the style of Ferber–McKinley–Samotij, and Theorem 1.7, proved by a more involved partite-shrinking argument that gives two bounds depending on whether f ≥ h−β or f < h−β. These are applied to generalized Turán numbers, planar generalized Turán numbers, and extremal problems for subsets of integers with forbidden/encouraged configurations. The paper is well motivated and carefully written, with explicit discussion of the necessity of hypotheses such as the no-isolated-vertices condition.","tokens_in":21956,"tokens_out":9250,"duration_ms":78505,"significance":"If the technical results are fully correct, the paper makes a substantial contribution: Theorem 1.5 gives a clean, general supersaturation bound with a short proof, and the partite-shrinking Proposition 4.5 is an interesting new technique. The applications to essentially arbitrary H and F, and to additive-combinatorial configurations, go beyond previous sporadic results. The paper also honestly identifies necessary conditions and gives counterexamples when they fail. However, the proof of the second bound of Theorem 1.7 contains a specific gap that is load-bearing: as written, it does not establish Proposition 4.8's second alternative, and hence Theorems 1.1(iii), 1.3, and 1.4(iii) are not proved. The first bound of Theorem 1.7 and Theorem 1.5 appear sound.","major_comments":[{"comment":"The proof asserts: 'if |I_t|<|t| then |I_t|≤|t|−2 since such a t has some t_i≥2.' This is false. For example, in a 3-partite H with |V'_1|=1 and |V'_2|,|V'_3|≥2, a type t=(1,1,0) has |t|=2 and |I_t|=1, with no t_i≥2. Such a type is compatible with the no-containment hypothesis: an H'-edge can use a vertex in V'_2 different from the vertex used by the F-edge. For |I_t|=|t|−1, the bound (7) gives only (C^{-1}k)^{1/(h−β−|t|+1)}, which is larger than k^{-2/(h−β−|t|+2)} for large k; hence the assumed upper bound e_t(F)≤c k^{2/(h−β−|t|+2)} does not imply (6). Consequently the proof does not establish e_t(F')=0, so Proposition 4.8's second bound and the second part of Theorem 1.7 are unproved. This also affects Theorem 1.1(iii), Theorem 1.3, and Theorem 1.4(iii).","section":"Section 4, Claim 4.10"},{"comment":"In Claim 4.7 the proof uses the inequality 'C>(20)^{4h^3}' to absorb the factor (20)^{j(h−s)}. The stated hypothesis of Proposition 4.5 only gives k≥C≥(30)^{4h^2}. For h≥2, (30)^{4h^2}<(20)^{4h^3}, so this inequality does not follow. This is repairable by enlarging the constant in Proposition 4.5 or by choosing C in Proposition 4.8 accordingly, but as written the proof of Proposition 4.5 is incomplete.","section":"Section 4, proof of Proposition 4.5"}],"minor_comments":[{"comment":"In the paragraph before part (ii), '1 β=0' should read 'β=0'.","section":"Theorem 1.1 statement"},{"comment":"The phrase 'the h-partite result' is unclear; partite language is defined for hypergraphs in Section 4, while the planar graph context here may confuse the reader.","section":"Proof of Theorem 1.3"},{"comment":"Typo: 'there must exists some v' should be 'there must exist some v'.","section":"Claim 4.3"},{"comment":"The displayed definition of C is missing a closing brace: it should be C = max{C' h^β (30)^{4h^2 β}, (30)^{4h^2}}.","section":"Proof of Proposition 4.8"}],"recommendation":"major_revision","confidential_remarks":"The paper has real merit and the main first-bound machinery appears correct, but the second bound of Theorem 1.7 is not established as written. The error in Claim 4.10 is not a typo; it is a missing case in the type analysis. I recommend major revision rather than rejection because the framework and the first main theorem are valuable, and the gap may be fixable with a more careful case distinction. The constant mismatch in Proposition 4.5 is also readily fixable but should be corrected. Please ensure the revised version addresses both issues explicitly."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Sam, quick take on Spiro's supersaturation paper. The headline: the hypergraph-weighted independent-set framework is genuinely useful, and the partite-shrinking machinery is original, but the proof of the second bound in Theorem 1.7 has a gap that undermines the f<h−β applications (Theorems 1.1(iii), 1.3(ii), 1.4(iii)) as written.\n\nWhat's new and good: The α_H(F) formulation is a clean way to unify graph and additive-combinatorics supersaturation. Corollaries 1.6 and the first bound of 1.7 (f≥h−β) are proved by a solid random-shrinking argument; the bounds are tight in the extremal case and recover Jiang–Longbrake and Fox–Pohoata. The applications to generalized planar Turán numbers and to systems of equations are clearly motivated. The paper is self-contained and the exposition is good.\n\nThe soft spot: Claim 4.10 asserts that if |I_t|<|t| then |I_t|≤|t|−2 because some t_i≥2. That's false. A type t=(1,1,0) in a 3-partite H with the first part not shrunk to a singleton has |t|=2, |I_t|=1, and no coordinate ≥2. Such a type is consistent with the no-containment hypothesis: an H-edge can use a different vertex in the non-singleton part, so it need not contain the F-edge. Repeating the estimate (7) for this case gives k^{-1/(h−β−|t|+1)}, which is too weak to force e_t(F')<1. So the second bound of Theorem 1.7 is not proved as written. This is not a small cosmetic gap; it removes the proof of the f<h−β cases the paper advertises. It may be repairable—for instance, the variant in §5 with |I_t|≤r suggests the author knows the right parameter—but the current text doesn't go through.\n\nAlso worth checking: the 'first ever' claim in §1.1 isn't reconciled with cited [13] (Gerbner–Nagy–Vizer), which the reader flagged. I'd ask the author to clarify what Theorem 1.1 adds over that paper. And the no-isolated-vertices condition on F is load-bearing; the author supplies a counterexample when it fails, which is honest, but it should be stated as a hypothesis in Theorem 1.1's headline.\n\nNet: the paper deserves peer review—the framework and first bound are strong and worth publishing—but the author needs to fix the Claim 4.10 gap before the small-F results can be accepted. I'd send it to a serious referee.","headline":"Strong framework and first bound, but the second bound of Theorem 1.7 rests on a false step in Claim 4.10; the paper needs a fix before the small-F applications are usable.","tokens_in":22451,"tokens_out":7693,"would_cite":true,"duration_ms":68062,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C35","05C65","05D40"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper establishes a universal supersaturation theorem: for any hereditary family of hypergraph pairs with extremal function O(n^β), having k n^β edges of H forces Ω(k^{f/(h−β)}) edges of F whenever f≥h−β.","keywords":["supersaturation","generalized Turán numbers","hypergraph pairs","weighted independent sets","hereditary families","extremal combinatorics","systems of linear equations","partite hypergraphs"],"falsifier":"Take H=F∪K_1 and let G be a copy of F together with k isolated vertices. Then #(H,G)=k while #(F,G)=1, and since ex(n,H,F)=0 for sufficiently large n, the condition #(H,G)≥k ex(n,H,F) holds trivially; this concrete example settles that any universal supersaturation statement of Theorem 1.1 fails if the no-isolated-vertices assumption is dropped.","tokens_in":21453,"feed_emoji":"🧮","tokens_out":9567,"duration_ms":85278,"temperature":0.7,"pith_summary":"The paper establishes a general supersaturation theorem: in any hereditary family of hypergraph pairs (H,F) whose extremal function grows at most like n^β, any member with k n^β edges of H must contain Ω(k^{f/(h−β)}) edges of F, where h and f bound the edge sizes of H and F. It reaches this by viewing extremal problems as weighted independent sets: independent sets of F are rewarded by how many H-edges they induce. If correct, this yields the first fully general supersaturation bounds for generalized Turán problems, meaning graphs with many copies of one subgraph H are forced to contain many copies of another subgraph F. The same machinery gives counterparts for planar graphs and for subsets of integers that avoid one system of linear equations while maximizing solutions to another. The payoff is a single explanatory exponent that transfers supersaturation arguments from edge counts to arbitrary subgraph counts.","feed_headline":"Many H-copies force many F-copies: exponent f/(h−β)","feed_subtitle":"General theorem covers graphs, planar graphs, and integer sets avoiding one system of linear equations.","key_machinery":"The central object is the hypergraph pair (H,F) on a common vertex set, with α_H(F)=max{e(H[I]): I independent in F}; 'large' independent sets are those inducing many H-edges. The proof's engine is a part-shrinking process (Proposition 4.5) applied after Lemma 4.1 extracts an h-partite subhypergraph of H with a constant fraction of its edges. For each part V_i, Lemma 4.2 supplies a subset V_i′ and a factor p_i so that H loses at most a factor ∏p_i of its edges, while F-edges of type t=(t_1,…,t_h) — the vector of intersection sizes with the parts — are bounded in the surviving subhypergraph by ∏p_i^{t_i} times their original count, up to a (40h)^{O(h²f′)} constant. Iterating at most 4h² times","core_discovery":"The central claim is that supersaturation for a broad class of extremal problems is governed by one ratio, f/(h−β). Concretely, let P be a hereditary family of hypergraph pairs (H,F) in which H-edges have size at most h and F-edges have size between f and f′, and suppose the extremal value ex(n,P), the maximum H-edge count over n-vertex F-independent objects, grows at most like n^β. Theorem 1.7 asserts that if f≥h−β, any (H,F)∈P_n with e(H)≥k n^β must have e(F)=Ω(k^{f/(h−β)}); when f<h−β, the same conclusion holds in the weaker form Ω(k^{2/(h−β−min{f′,⌈h−β⌉−1}+2)}) provided no edge of any F is contained in an edge of any H. The proof reduces to partite subhypergraphs and shrinks each part by","pith_inferences":["The exponent f/(h−β) looks like a dimension: h is the ambient edge size and h−β is the codimension of the extremal family. If this reading is right, the same formula should predict supersaturation exponents for other hereditary settings with ex(n,P)=O(n^β), including hypergraph Turán problems, before a dedicated proof is found.","The paper's second-case bound is self-consciously an artifact of the method, and its Question 5.1 asks whether Ω(k^{f/(h−β)}) can hold even when f<h−β under stronger intersection conditions; resolving that question would tell whether the two regimes are genuinely different.","The hypergraph-pair formalism points toward counting: supersaturation theorems are the standard input for counting results and container-type arguments, so one natural next step is to use these bounds to count the number of large H-rich, F-poor sets, in both the graph and integer settings.","For systems of linear equations, the type decomposition suggests the framework extends to configurations with unequal variable roles, not just uniform arithmetic progressions, as long as solutions are encoded as hyperedges of size between f and f′."],"forward_implications":["Graphs: if #(H,G)≥k n^β and ex(n,H,F)=O(n^β), then #(F,G)=Ω(k^{f/(h−β)}) when f≥h−β, and Ω(k^{2/(h−β−f+2)}) when f<h−β and F is not a subgraph of H; in particular, graphs with C′ex(n,H,F) copies of H have at least C copies of F.","Planar graphs satisfy the same two exponents, so n^ε·ex_P(n,H,F) copies of H force n^δ copies of F for some δ>0.","Sets of integers: a set A of size n with #_N(H,A)≥k n^β must contain Ω(k^{f/(h−β)}) solutions from F when F-solutions have size at least h−β, giving a common framework for problems about arithmetic progressions and Sidon-type equations.","A deletion-based general bound gives Ω(k^{(f+h−1−β)/(h−β)} n^{β−h+1}) in all hereditary bounded pairs, and this bound is best possible in the extremal case e(H)=Θ(n^h) with f-uniform F."],"fun_headline_variants":["Supersaturation: H-edges force F-edges at rate f/(h−β)","Many H-copies force many F-copies: exponent f/(h−β)","H-weighted independent sets: supersaturation exponent f/(h−β)","General supersaturation: one ratio f/(h−β) for Turán and equations","Hypergraph independent sets: supersaturation via f/(h−β)"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing structural premise is that the forbidden object F has no isolated vertices (equivalently, in the small-parameter regime, that no F-edge sits inside an H-edge); without it the forced-count conclusion is simply false, as the paper's own counterexample shows.","fun_headline_variants_meta":{"raw":{"variants":["Supersaturation: H-edges force F-edges at rate f/(h−β)","Many H-copies force many F-copies: exponent f/(h−β)","H-weighted independent sets: supersaturation exponent f/(h−β)","General supersaturation: one ratio f/(h−β) for Turán and equations","Hypergraph independent sets: supersaturation via f/(h−β)"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000794,"raw_usage":{"total_tokens":3332,"prompt_tokens":745,"completion_tokens":2587,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":489,"completion_tokens_details":{"reasoning_tokens":2484}},"tokens_in":489,"tokens_out":2587,"duration_ms":19129,"temperature":1.0,"reasoning_tokens":2484,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T02:58:47.186939+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take H=F∪K_1 and let G be a copy of F together with k isolated vertices. Then #(H,G)=k while #(F,G)=1, and since ex(n,H,F)=0 for sufficiently large n, the condition #(H,G)≥k ex(n,H,F) holds trivially; this concrete example settles that any universal supersaturation statement of Theorem 1.1 fails if the no-isolated-vertices assumption is dropped.","supporting_citations":[],"review_version":1}